Grammar Systems - A Language-Theoretic Approach to Distribution and
Cooperation

by Erzsébet Csuhaj-Varjú

Grammar systems' is a recent vivid field of formal language theory
modelling distributed complex systems. The theory provides highly elaborated
frameworks and tools for describing various kinds of multi-agent systems
at the symbolic level, from distributed and cooperative problem solving
systems to artificial life systems, collective robotics, mentioning only
a few. It can be used in DNA computing as well. The area has been intensively
investigated by several cooperating research groups from about a dozen of
countries (Europe, Canada and Japan) for years.

In present day computer science, artificial intelligence, cognitive psychology
and in other related fields we have to deal with complex tasks distributed
among a set of agents (processors) which work/live together in some well-defined
manner. Parallel computers, computer networks, distributed databases, and
knowledge sources are practical materializations of this idea. Similarly,
psychologists speak about the modularity of mind, in problem solving theories
many models based on cognitive agents' cooperation appear.

Since formal language theory is a well developed theoretical framework
for modelling aspects the essence of which can be captured at the level
of symbol systems, a challenge appears: how to describe (the behaviour of)
multi-agent (complex distributed) symbol systems in terms of language identifying
devices. Grammar systems theory attempts to answer this challenge. Roughly
speaking, a grammar system consists of several grammars (automata, or other
language identifying mechanisms) that cooperate according to some well-defined
protocol in deriving sentential forms of a language (or languages). The
components of the system correspond to the agents, the current string(s)
in generation to a symbolic environment, and the system's behaviour is represented
by the language or the string sequences identifying the current state of
the system. In addition to distribution, cooperation, communi-cation, other
important notions as emergent behaviour can also be formalized in this context.

We illustrate the great variety of models offered by the theory through
the most important frameworks:

The investigations started in 1988 by introducing cooperating/distributed
grammar systems for modelling the syntactic aspects of the blackboard model
of problem solving. In this case the cooperating independent agents are
represented by generative grammars which, under some cooperation strategy,
derive in turn a common sentential form (the blackboard) in order to generate
a common language. The achieved results demonstrate the power of cooperation
showing that systems with syntactically very simple components under some
appropriate cooperation strategy are able to generate complicated languages
of very powerful language classes. Interesting results are, among other
things, that any computation under some kind of competence-based cooperation
of grammars can be performed in the same kind of system with three co-operating
partners. Computation under hybrid (different) cooperation strategies does
not need more than four grammars cooperating to determine the same language.

Team grammar systems with simultaneous actions of some grammars in the
system (teams) which cooperate in deriving a common sentential form, demonstrate
an equivalence between programming the sequence of actions and computation
under some kind of competence-based cooperation of freely chosen grammar
teams with a very limited number of components.

Colonies, motivated by subsumption architectures of R. Brooks, describe
language classes in terms of behaviour of collections of simple, purely
reactive, situated agents with emergent behaviour. In this model the agents
are represented by very simple regular grammars. The basic variant characterizes
the context-free language class, while the more sophisticated models (competition
among the agents, timing, etc.) lead to considerably enhanced descriptive
power.

Eco-grammar systems are grammatical models of ecosystems: developing
grammatical agents in a (dynamically changing) population interact with
each other and with their shared evolving symbolic environment. The framework
provides tools for describing life-like phenomena (birth, death, hybernation,
overpopulation, pollution, etc.) in terms of formal grammars and languages.

Networks of language processors form an essential part of the area. In
this case each node of a virtual graph is represented by a language processor
(a language identifying device). Each processor works on a string (on a
collection of strings) and informs the others about its activity by communicating
strings which are data and/or programs. Rewriting and communication take
place alternately, the system functions (usually) in a synchronized manner.
Parallel communicating grammar systems, a highly elaborated field, with
Chomsky and Lindenmayer grammars at the nodes, studies networks with components
communicating data strings by request. Test tube distributed systems based
on splicing and that of cutting and recombination are particular cases of
the model with components using variants of DNA recombination and they realize
computationally complete and universal machines (in some cases with a limited
number of components). Investigations have been started for implementing
ideas of the Wave paradigm of active knowledge networks in this framework.

The results in the above areas address mainly theoretical problems but
the research community is open to any suggestions for applications.